Paper to be presented at the

35th DRUID Celebration Conference 2013, Barcelona, Spain, June 17-19

Problem-solving and Generality as Sources of Growth and HeterogeneitySimone Vannuccini

Friedrich-Schiller-Universität JenaFaculty of Economics and Business Administration

simone.vannuccini@uni-jena.de

AbstractThe basic idea of this paper is that heterogeneity arises in a market because firms, meant as problem-solving entities,develop specific knowledge to rule out the bottlenecks and the technical challenges they have to face. The newlycreated knowledge is embodied in an innovation, and therefore an alternative explanation of the cause of innovativeactivity at the firm level can be found in the necessity to overcome problems, more than in the reaction to opportunitiesand incentives. Moreover, problem-solving activities can end up in different types of innovations, characterized bydifferent "degrees of generality", which in turn affect firms' productivity. Fluctuations in productivity can therefore be theemergent result of many co-existing technological (capital vintages) dynamics. The setting of a simple model is finallyproposed in order to summarize the theoretical building blocks introduced in the paper.

Jelcodes:O31,-

Problem–solving and Generality as Sources of Growth

and Heterogeneity

February 28, 2013

Abstract

PRELIMINARY DRAFT - PLEASE DO NOT CITE. The basic idea of this paperis that heterogeneity arises in a market because firms, meant as problem–solvingentities, develop specific knowledge to rule out the bottlenecks and the technicalchallenges they have to face. The newly created knowledge is embodied in an inno-vation, and therefore an alternative explanation of the cause of innovative activityat the firm level can be found in the necessity to overcome problems, more than inthe reaction to opportunities and incentives. Moreover, problem–solving activitiescan end up in different types of innovations, characterized by different “degrees ofgenerality”, which in turn affect firms’ productivity. Fluctuations in productivity cantherefore be the emergent result of many co–existing technological (capital vintages)dynamics. The setting of a simple model is finally proposed in order to summarizethe theoretical building blocks introduced in the paper.

1 Introduction

The empirical research in the field of industrial dynamics has highlighted a rich statis-tical structure holding along the evolution of industries, nicely fitting distributions offirm’s size, profitability, and productivity (Dosi, 2007). The persistency feature of theregularities seems to rely on complex-phenomena mechanisms (Kirman, 2010), wherethe statistical patterns are emergent properties, broken symmetries unfolding from theaggregation of micro behaviors. Stable structures at the industrial and macro level de-velop from a deep heterogeneous micro scenario, where the variability and the disper-sion of individual characteristics doesn’t decline the more disaggregate the unit of analy-sis are. In addition to the heterogeneous distribution of characteristics and competenciesamong economic agents, the “shape”, the pace and the rate of the dynamics in an in-dustry strongly depend on its degree of turbulence (Cantner, 2011), that is the regime ofentry, exit and selection prevailing in a given period.

Market turbulence and individual heterogeneity affect each other, turbulence increas-ing the incentives and the constraints for firms to adopt specific behaviors and tech-niques, heterogeneity influencing the degree of fitness and success in interaction, meantboth as economic competition and cooperation. The process outlined is evolutionary innature1, thus suggesting that an explanation to relevant research puzzles and paradoxessuch as “why firms differ” (Nelson, 1991) or “why growth rates differ” (Fagerberg, 1988)

1This claim does not necessarily imply the generalization of Darwinian categories to economic phe-nomena; for the theoretical debate about the meaning of evolution and Darwinism in economics see, forexample, Witt and Cordes (2007), Vromen (2010), Buenstorf (2006).

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at the macro level has to be found in the sphere of economic dynamics. Turbulence, thatis a form of restless change (Metcalfe et al., 2006), together with the emerging hetero-geneity of economic actors, are the results of a process of differential growth, which inturn is strongly depending on the outcomes of the innovative activities undertaken byfirms (Cohen and Levin, 1989).

The centrality of innovative activity as a determinant of differential growth is at themoment a consensus amongst appreciative theorists, but the forces acting behind thedecisions to engage in innovative activities are still a matter of debate. As a result ofthe inconclusive inquire into the role of proximate causes such as firms’ size and marketstructure (Cohen, 2010), the neo–Schumpeterian literature stresses two main reasons forinnovative activity to occur, namely the exploitation of technological opportunities andthe potential returns coming from economic chances (with particular attention to de-mand conditions), to which it should be added the inducement argument. In the lattercase, innovative behavior follows the changes in factor prices or — in a more systematicapproach (Antonelli, 2007) — is fostered by the mismatch between firm plans and actualconditions. The exploitation of technological opportunities relies both on economic andtechnical arguments to explain innovative behaviors, and captures nicely the stylizedfact of a rich technological variety and dynamism of industries; nonetheless, specifi-cation and measurement difficulties on the empirical side (Levin and Reiss, 1984) andmodeling complexities (Cantner and Pyka, 1998) have constrained its adoption outsidethe field of innovation studies.

The pure economic explanation of innovation incentives, that stemming from profit–maximizing behaviors, both in industrial organization non–tournament models (Das-gupta and Stiglitz, 1980) and in incumbent-entrants races (Reinganum, 1985), is by farthe most relevant in the literature. The same holds for the modeling of technologicalchange in macroeconomic (endogenous) growth models approaching steady–state equi-libria. Two assumptions characterize these modeling strategies. The first concerns theeconomic incentive to reallocate resources to innovative activity (or, more in general,to Research and Development), which derives from the inter–temporal optimization ofmarginal returns in different activities or from the decision to participate in a race wherethe benefits of the monopoly right after an innovation are challenged by the threat of po-tential entrants in the market. The second assumption has to do with knowledge, whichis usually considered to be additive and homogeneous. Knowledge generation and accu-mulation, in particular for what concerns growth models, are seen as the main forces atwork behind innovation and growth in productivity, variety and quality; unfortunately,its characterization is far from being satisficing.

In this paper, I attempt to tackle several of the issues mentioned above. First, con-cerning the mechanisms leading heterogeneity to arise among economic actors, I suggestthat an alternative explanation for innovative activity can be found looking at firms asproblem–solving entities. From this perspective, innovations are meant as the outcomeof a process of solutions search in order to solve technical and production problems.Innovations cannot be just an outcome of the optimal allocation of resources betweenalternative ends, if firms need them in order to survive in the market. Connected tothis point is the dynamic of knowledges production. Conversely to the representationof knowledge in growth models, that of a generic stock uniformly affecting the econ-omy, knowledge is inherently “directed”, specific to firms and applications (Pavitt, 1984)and localized to deal with problems and with the search for solutions. The iterated pro-cess of “taylor–made” (Aghion and Howitt, 1998) knowledge production aimed to solvefirm–specific problems is the driver of heterogeneity in this paper.

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The second contribution of the paper comes as a corollary of the first point. Theoutcomes of the problem–solving activity in terms of knowledge production and tech-nological innovation will affect firm’s performances, and more precisely productivityand profits. In particular, a micro–level explanation of productivity’s dynamics will besuggested based on a peculiar characterization of the process of technological change. Inshort, what is introduced is a differential impact of each innovation, in turn determinedby the innovation’s breadth of applicability, interpreted as the capability to solve largeror smaller sets of problems. This is realized via the introduction of a innovation–specificdegree of generality, in turn inspired by the literature on General Purpose Technologies(GPTs, from now on).

In order to combine the building blocks mentioned in the paper, I will introduce thesetting of a simple model, non–radical in its structure, but novel in its premises and aims.A final remark concerns the scale of analysis; the paper talks about firm’s behavior, but ismeant to represent micro-level relations capable to produce aggregate results. Therefore,I will refer both to growth and industrial dynamics literature, aiming at bridging thosetwo research traditions.

The paper proceeds as follows. Section 2 deepens the discussion about the natureof knowledge, its representation and its role in directing the behavior of firms. Section3 introduces the other building block of the analysis, namely the concepts of problem–solving and generality. Section 4 outlines the structure of the model to summarize thecentral tenets of the paper and to show their interaction. Final section concludes.

2 On the Directedness of Knowledge

The basic idea of this paper is that heterogeneity arises in a market because firms, meantas problem–solving entities, develop specific knowledge to rule out the bottlenecks andthe technical challenges they have to face. Usually, the newly created knowledge is em-bodied in an innovation, and therefore an alternative explanation of the cause of innova-tive activity at the firm level can be found in the necessity to overcome problems, morethan in the reaction to opportunities and incentives. In order to develop this argument,the claim that knowledge is largely firm specific has to be — at least briefly — defended.

Most of the literature accepts Arrow’s (1962a) interpretation of knowledge (or, in Ar-row’s later terms, technical information (Arrow, 1996)) as a peculiar commodity, whosepublic good nature affects welfare and social optima. Conversely, the very nature ofknowledge is much more complex and multifaceted2. First, knowledge has a tacit com-ponent difficult to replicate, to embody or to codify. However, the share of knowledgethat is structurally non-codifiable seems to be small and therefore not so relevant (Cowanet al., 2000) to challenge the Arrow’s legacy. Second, the availability and the diffusion ofknowledge can be reduced or slowed by appropriation, i.e. measures of knowledgeprotection and patenting, conceivable as devices to collapse the public good features ofknowledge into that of a latent public good. Third, knowledge is fragmented and dis-persed between a high number of agents, each of them endowed with limited computa-tional capabilities and able to perform mainly local search in the knowledge and technol-ogy space (Nelson and Winter, 1982). Fourth and central for this paper, knowledge canbe sticky and difficult to exchange or replicate because it is produced to fulfill specific

2I don’t tackle here the issue of knowledge definition (Mokyr, 2005; Saviotti, 2007), its externalist orinternalist (cognitive) nature, its relation with concepts such as data and information (Boisot and Canals,2004) or correlated understanding (Metcalfe and Ramlogan, 2005) and the forces governing its evolution(Olsson, 2001; Zambelli, 2004; Weitzman, 1998).

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purposes. Knowledge is not a homogenous set or structure (Olsson, 2000; Saviotti, 2007),nor it is by definition overlapping across firms and industries; instead, it is directed. AsPavitt puts it: “most technology is specific, complex. . . [and] cumulative in its development. . . Itis specific to firms where most technological activity is carried out, and it is specific to prod-ucts and processes [. . . ]” (Dosi and Nelson, 2010). As a consequence, the exploitation ofknowledge externalities becomes a less trivial process, requiring an adequate absorptivecapacity (Cohen and Levinthal, 1989) that firms have to actively develop.

An explicit treatment of knowledge is also the most distinctive character of the so–called endogenous and Schumpeterian literature on economic growth (Olsson, 2001).However, this focus on knowledge, probably driven by the need to enquire the veryultimate causes of growth during the worldwide transition from a pure manufacturing-based to a knowledge–based economy, casted shadows more than shed lights: the lackof precision in the definitions of what knowledge is or should be for growth theory, theperfect mapping between knowledge and technology, the puzzling mix between cardi-nality and ordinality in representing the state, the level or the stock of knowledge, themeasurement limits, all led to a “depressing” confusion (Steedman, 2003).

From an analytical point of view, this confusion translates into the use of differen-tial equation to explain the law of motion of knowledge, assuming in most of the casesa linear relation — a knowledge production function — between the growth rate of theknowledge stock and the number of researchers employed in a stylized R&D sector. Sub-ject to knife-edge assumptions and scale-effects (Jones, 1999), a number of refinementsto the modeling of knowledge have been suggested in the literature (Dinopoulos andSener, 2007), unfortunately without modifying the treatment of knowledge as a generic,additive and usually disembodied cardinal stock of resources to accumulate. The ana-lytical advantages to aggregate knowledge in a single scalar are hardly compatible witha picture of heterogeneity emerging from firms’ specific knowledge endowments.

The abstract propositions, prescriptions and procedures borrowed from Science ordeveloped within firms in response to idiosyncratic needs or problems, are realizedthrough, or encoded in, technological artifacts and devices (and also in non–technologicalones, such as organizational schemes, routines, institutions, practices). The process ofencoding, in turn, guarantees a potentially costless reuse of knowledge (Langlois, 2001),creating the conditions for increasing returns and non–convexities in production to ap-pear. Via the development of technological artifacts or the direct encoding in them,the evolution of knowledge affects differential growth dynamics while remaining in thebackground. For modeling purposes, this means that knowledge has not to be explicit —as it is instead in endogenous growth models —. Its embodiment in technological com-ponents first and then in firms’ capital structure is therefore a realistic modeling choice.Arrow’s learning by doing model (1962b) is the pioneering example in this sense, to-gether with the kaldorian tradition based on technical progress functions (Lorentz andLlerena, 2004), the evolutionary quasi–vintage growth models such as Silverberg andLehnert (1993) and Silverberg and Verspagen (1994) and the recent reprise of kaldorianarguments in the adaptive economic growth approach (Metcalfe et al., 2006; Foster andMetcalfe, 2010). The present paper aims to give an additional contribution in this trajec-tory.

3 On Problems, Solutions, and Generality

To adopt an interpretation of knowledge as a firm specific resource that is encoded intechnologies and embodied in capital structures constitutes the paper’s first theoretical

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building block. Some additional words should now be spent on two related issues: onthe one hand, the determinants of the production of new knowledge and, on the other,the characteristics of this new embodied knowledge. Given that innovative activity isa form of knowledge production, what I will eventually propose is an explanation ofthe firm’s uneven propensity to commit to innovation, a propensity dependent in turnon a process — the emergence of problems and the search for solutions — that is onlypartially endogenous to the firm.

Concerning the determinants, already in the introduction it was suggested that theneed to solve specific problems a firm (and, in general, an economic agent) face is thedriver of knowledge production and, accordingly, innovation. Humans and organiza-tions are problem–solvers, making use of procedural rationality (Dosi and Egidi, 1991),heuristics, routines and algorithms to decompose and complete their tasks. Productionprocesses also can be represented — from an engineering perspective — as sequencesof task or problems (Kremer, 1993; Buenstorf, 2005; Lombardi, 2009) to be fixed. Alsoin the evolution of technological systems (Arthur and Polak, 2006), as it happens in hu-man problem solving (Newell et al., 1958), the logic governing progress and learning isthat of an iterative succession of duple problem–solution, where every new completedstep (a solution found, a new “piece” of knowledge developed) serves as a module oran executable (Arthur, 2009) to be used hierarchically or sequentially for the process tocontinue.

The need to tackle a precise and defined (sometime ill–defined) bottleneck constraintsthe range of choices for economic agents and thus imposes a directed (even if not irre-versible) structure on the process of change. In sum, and abstracting from other possibledeterminants, firms become heterogeneous not just because of their initial conditionsand expertise (Klepper, 1996), as if they are “born” heterogeneous, but also due to thefact that a unique unfolding process of specific problem–solving activities defines thepath and the trajectory of their evolution. Since the development of problem–solvingknowledge, exemplified by firms’ innovative activities, is actively influenced by the de-cisions firms take, we cannot treat the problem–solving perspective as a deterministicone. Problems and bottlenecks trace a forced direction, but the choice of certain innova-tions as solutions between alternatives (combined with the past history of each agent)has the potential to continuously open new opportunities and trajectories. However, ateach point in time, a firm capital structure — representing its technological architectureas emerged through time and as a result of sequences of problem–solving activities —tells us the story (at least part of it) of a localized technical change made in response notonly to economic incentives opportunities, or inducement by factor prices.

The second issue to deal with is about the properties characterizing the problem–solving knowledge and innovations created and adopted by firms. What makes firmsheterogeneous is the fact that the innovations they adopt are heterogeneous themselves.Here it is interesting to measure this dispersion, or variety, in terms of generality. Startingwith a quote from Herbert Simon, stating that “no single-purpose device is going to bringabout a revolution, however convenient or useful it may be. Revolutionary significance lies ingenerality” (Simon, 1987), it is useful to define the concept of generality as the breadth ofpurposes to which a technology could be applied. In turn, this breadth of purposes is areflection of the range and types of different problems a technology can solve. A moregeneral technology, embodying a more general problem–solving knowledge, will alsohave a stronger impact on the technologies complementary to it.

The main idea here is to introduce a “degree of generality”. Each innovation belongsto a continuum of technologies, whose scale is defined by the capability to solve prob-

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lems. On the one end of the continuum there are technologies so specific to be able tosolve just singular problems, hence pursuing a limited set of functions. On the oppositeend one can identify technologies that are very general, introduced in the market for spe-cific reasons but then capable to solve a wider range of problems. Innovations such asthe electric dynamo, the laser or semiconductors, initially invented or exploited for spe-cific aims, have later shown to be capable of successfully tackle a much more extensiveset of problems.

The recognition that technologies are not all equal is shared with the literature onGPTs (Bresnahan and Trajtenberg, 1995; Helpman, 1998; Carlaw and Lipsey, 2006). GPTsare defined as peculiar technologies that, performing some “generic functions”, display ageneral applicability, are capable of ongoing technical improvements and foster comple-mentary innovations in application sectors (Rosenberg and Trajtenberg, 2004). The defi-nition and modeling of GPTs suffer from two limitations: the first concerns the ex–anteidentification, in the sense that before its diffusion (a process that can last for decades)it is impossible to decide if a technology is general–purpose or not; the second is theproblem of classification. Once a technology has been recognized as “important” or rev-olutionary, it can just fit into one out of two categories: GPT or non–GPT; which shouldbe in this case the empirical criteria to distinguish between what is a GPT and what isnot? Given this theoretical shortcomings, GPT–based models (ranging again from indus-trial economics to growth theory) overcame the puzzle assuming that a GPT is alreadyknown to be general–purpose at the moment of its arrival, both in deterministic and instochastic settings. Except in a case, that of the model of van Zon et al. (2003), the gen-eral character of a technology never emerge endogenously, because it is already decidedfrom the setup of the models.

The rationale for avoiding the modeling of ex–ante uncertainty about the GPT–natureof a technology is best explained by analytical difficulties. Instead, the decision to inventa new technology — already known to have GPT characteristics — and to invest in it ismade in the literature dependent on a double externality, horizontal as well vertical. Thelatter is based on a dual inducement mechanism, in the sense that the revenue functionsof both GPT and application sectors are interdependent: investing more in the GPT in-creases the returns to invest in complementary technologies, and the other way round.The former type of externality is rather caused by the number of application sectors towhich the GPT can be applied. The general applicability part of the GPT definition, oftensubstituted by the weaker formulation “widely used” (Bresnahan and Yin, 2010), refersmostly to the last point: general is meant in terms of diffusion and adoption across alarge number of industries and sectors.

The interpretation of generality used in this paper is different with respect to thatprevailing in the literature, and it is more coherent with the specification of GPTs as car-riers of generic functions (Rosenberg and Trajtenberg, 2004). Moreover, the classificationproblem is overcome here, since to infinitely different problem–solving capabilities ofa technology corresponds an equally infinite range of degrees of generality. Defininggenerality according to the problems an innovation is able to solve helps us also to un-derstand why, despite the prediction of low–level equilibria and no–investment trapssuggested by incentive–based models, important innovations actually occur and diffusein the economic system.

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4 Problem–solving and Generality at Work, in a Simple Model

Taking stock of the discussion developed so far, a basic model it is now introduced,aimed to show the interaction between firms’ problem–solving activities, innovative ac-tivities, and performance in term of productivity and profits. The model is meant to bean initial benchmark, where a single profit-maximizing firm deals with decisions aboutthe R&D investments, which in turn affect both the problem-solving success and thefluctuations in productivity.

The model is a pseudo–vintage model3 constructed ad hoc (but inspired by the argu-ments of Nelson and Winter (1982), Schuette (1994) and Silverberg and Verspagen (1994))to focus on the central tenets highlighted so far. The main conceptual difference with re-spect to the evolutionary literature is that vintages are not competing techniques, subjectto selection via a traverse and according to the profits they generate (Nelson and Winter,1982, Ch. 10). Instead, they are technological components of a “compositional” work-ing architecture (Lombardi, 2009). Therefore, even if the firm-level production functionanalytically permits a certain degree of substitution between the vintages in use, the the-oretical production scheme assumed here is that of short–run quasi–fixed proportionsmodel, where complementarities are endemic (Dosi and Grazzi, 2006).

To center the analysis on the structural relation between problem–solving and firmperformances, demand is ruled out from the model, so that the market is competitive andalways clearing. Also the financial and credit market exists and is perfect; firms can justborrow — at a negligible interest rate — the resources needed to engage in investments.A representative firm i produces at time t an output qi,t equivalent to its revenues withthe linear additive production function

(1) qi,t = ηit

M

∑m=1

ki,m,t.

In what follows, the index i is dropped to ease the reading. The variable ηt representsthe aggregate productivity of the firm, km,t is a capital vintage and m = 1, . . . , M indexesthe number of vintages a firm has in use. Concerning labor requirements, in order tokeep the model simple, I assume that to each vintage corresponds a worker, so that thenumber of workers in the firm is equal to Mt. In each period, the firm incurs a problemthat has to be solved. A failure in problem–solving will generate an additional cost F,whose magnitude can cause the firm to have negative profit and eventually to exit themarket. The problem is represented by the absolute value of a real number pt, extractedrandomly from a Normal distribution with mean zero. I avoid in the model a specifica-tion of the qualitative type of the problem4. In order to let the nature of the problem tobe affected by firm’s characteristics, the variance of the distribution is a function of thefirm’s size, proxied by the number of vintages in use (that corresponds also to the num-

ber of workers hired), such that σ2 =(

1 − 1Mt−1

)

. Larger firms will then have “bigger”

problems, as a burden deriving from their complex structure.The occurrence of problems cannot be avoided, following the realistic argument that

3The use of the word pseudo here is meant to distinguish the use of a generic concept of capital vintage,as in the present model, with respect to the vintage-based growth models in the neo-classical and endoge-nous tradition, whose principal aim is to determine the optimal scrapping time for different components ofthe capital stock (Boucekkine et al., 2011).

4See for example Cohen et al. (1972) for a different approach (the so–called garbage can model), aimedto explain organizations’ structures and hierarchies more than innovative activities and economic perfor-mances.

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firms have to continuously deal with bottlenecks and challenges in order to prolongtheir stay in the market. To impose that, the variance of the distribution must be pos-itive. Therefore, in the model firms enter the market with a minimum of two capitalvintages. The emergence of problems introduces strong uncertainty; I assume that theproblems distribution is unknown to the agents, preventing them to calculate expectedvalues before the actual arrival of a problem.

Once a problem has emerged, the firm searches for a solution. A solution st canbe found or not; the failure in solving the problem decreases the survival probabilityof the firm. Search is done by investing in R&D, with the aim to develop a specificproblem–solving knowledge. As already mentioned, R&D resources can be borrowedon the market without the need to invest in them a share of the profits (which is, instead,the modeling strategy chosen by models such as Nelson and Winter (1982) or Fatas–Villafranca et al. (2011)). The main point here concerns the decision about the amount ofresources to be invested in research. For a myopic firm, the R&D expenditure will be afunction of the expected effect generated on the production process by the new vintage,incorporating the problem solution. In fact, if a solution is found, it is embodied in a newcapital component, numbered Mt. This new vintage is equipped with a specific produc-tivity coefficient ηm,t, whose initial value is determined exogenously and it is commonfor all the vintages (so it can be set equal to one), but whose law of motion will differacross vintages.

The search process for solutions involves again the use of a distribution, this time aNormal centered on the value of the problem (µ = pt), whose variance is reduced accord-ing to the investment in R&D, which are characterized by decreasing returns. Formally,σ2 = ( 1

rdαt), with 0 < α < 1 and rdt representing research expenditure. In this sense,

R&D expenses can be interpreted as a sort of focusing device, useful to develop a solu-tion specifically directed to solve a problem. By shrinking the distribution around themean, firms attempt to match the precise value of the problem. While on the one handit can appear counterintuitive for firms to invest more in absolute terms in order to re-ceive very specific solutions instead of “broader” ones, on the other hand this behaviorreduces the risk connected with the failure of solving the problem, that is the occurrenceof high additional costs, capable to harm the firm’s production process.

The difference between the value of the problem and that of the solution (conditionedto its finding), so st − pt, determines the time–invariant degree of generality (gMt,with0 < gMt < 1) of the solution. Generality is meant here as the amount of problems acertain solution is able to solve. In turn, given that the solution (being specific knowl-edge) is encoded in a capital–embodied technological innovation, the degree of gener-ality comes to characterize the newly introduced capital vintage. In sum, according totheir problem–solving capability, different technological components of the firm’s pro-duction architecture will belong to different positions in the generality continuum. Atechnology characterized by a large positive st − pt will closely resemble a GPT, whilethe other way round holds for very specific small technological components. Obviously,a GPT cannot be fully characterized in this model, since the single firm’s behavior is forthe moment abstracted from industry and market interactions, excluding that diffusiondynamics take place.

What remains to be defined are the laws of motion for capital and productivity. Eachvintage accumulates according to the differential equation k = −δkm,t−1 + rt(g), whereδ is the rate of capital obsolescence common to all the vintages and rt(g) is a parame-ter which proxies the reconfiguration (or re–switching) that each technological compo-nent undergoes because of the arrival of the new vintage. In a very simplified way, the

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concept of re–switching (Arthur, 2009) is aimed to capture the architectural connectionbetween the technologies in a given production process: vintages are never scrapped,while the introduction of a new component induces them to a renewal of their struc-tural configuration, so to be “taylor–made” (Aghion and Howitt, 1998) and to fit withthe problem–solving innovation just introduced. Therefore, given the arrival of a newvintage, the value of each existing vintage increases in absolute terms if rt(g) > δkm,t−1.rt(g) is assumed to be an increasing function of g: the more general the new vintage, thebigger the reconfiguration imposed on the existing capital structure.

Firm’s productivity is defined as the capital-share weighted average of each vintage’sproductivity

(2) ηt =M

∑m=1

(

ηm,t

(

km,t

Kt

))

where Kt is the total capital stock. The law of motion governing the evolution of produc-tivity is η = −c(g) + ηg, where c(g) is an initial productivity slump (c(g) is set equal tozero after the first period, that of introduction in the firm) positively correlated with thedegree of generality, and the second term represents the increasing (but marginally de-creasing) dynamic contribution to productivity growth, that is smoother when the newvintage has a low degree of generality. As a consequence, a general technology wouldlargely contribute to productivity growth in the medium and long run, but it will inflicta bigger productivity slump in the very short run. The idea here is to impose in a simpli-fied version the so-called “time to sow and time to reap” cyclical dynamics produced bythe GPT–based growth models, so the productivity fluctuation due to the lagged effectproduced by the adoption of a new GPT. The central mechanism at work in the tradi-tional setting is one in which the adoption of a GPT will generate a productivity slumpin the beginning, followed by a major productivity rise. The size and the length of theslump is assumed equal for every cycle, or endogenously determined by the amount ofresearch employed in an intermediate sector producing GPT’s components. Even if theoccurrence of slowdowns is not detected for all the technologies, the expectation of fluc-tuations to happen is quite realistic, and has served as a convincing explanation of theSolow paradox (Solow, 1987; Basu et al., 2007).

The novelty introduced in this model is that fluctuations in productivity arise withineach vintage, while the firm’s aggregate productivity is the weighted sum of contribu-tion of all the vintages in use. Therefore, what happens at the firm level results from theinteraction of vintage–level states. For example, the initial slump introduced by a verygeneral technology could offset the positive contribution by many specific vintages. Isit better a small slump today or a big rise tomorrow? An “heroic” rational firm shouldbe able to calculate the level of productivity at every period, and to decide how much toinvest in research in order to find a problem–solving innovation with a degree of gen-erality capable to exactly contribute to productivity in a measure compatible with timepreferences and with the profit maximization problem. However, expectation can beformed only on the value of the solution obtained, while the distribution of the prob-lems is unknown to the agents: firm who can only act as a myopic maximizer.

Speculating on time preferences, firms more apt to prefer higher productivity levelstoday will invest more in R&D, reducing the risk of failure and also the initial slumps.However, they renounce to bigger gains in the future, as well as to the possibility to dis-cover very general technologies. This fact is in line with what happens in reality, whereapplied research rarely (or just “serendipitously”) develops revolutionary technologies,

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whose general principles and applications are usually discovered in Government–fundedor University labs.

Having information only about the current problem, myopic firms try to keep theproductivity level high by increasing R&D expenses. Eventually, the firm i solves itsmaximization problem

(3) maxE(π)t = qt − (wMt)− (r(g)t(Mt − 1))− F − rdt,

Where E(π)t are the expected profits at time t (if E(π)t < 0 the firm exits the market),wMt is the wage rate (assumed to be exogenously fixed by mechanisms such as publicdecisions or collective contracting) multiplied by the number of workers (correspondingto M, the number of vintages), rt(g)(Mt − 1) is the total reconfiguration cost of the ex-isting capital (last vintage excluded), F is the cost of the problems unsolved, with F = 0if s ≥ p, and F = ǫπt−1 (with 0 < ǫ < 1) if s < p; rdt is the investment in Research andDevelopment.

5 Conclusions

According to the industrial economics literature, firms engage in innovative activitiesbecause of their size or if induced by the structure of the market. Being unsuccessfulto confirm empirically the theoretical implications, industrial dynamics scholars intro-duced new industry–specific explanatory variables in the picture, such as technologi-cal opportunities, appropriability conditions, and the characteristics of the industry’stechnological–knowledge base. In this paper, I suggested that another track can be fol-lowed, that of interpreting innovative activities as an outcome of firm’s attempt to solvespecific problems. While developing solutions to the challenges continuously arising,firms build also their own specific knowledge endowment, which is cumulative and di-rected. As a consequence, heterogeneity grows fast across the agents.

The results of this problem–solving led innovative activities also influence the degreeof turbulence in the market: failing to innovate, firms risks to exit the market. In additionto that, turbulence is affected by the uneven and fluctuating firm’s productivity dynamic,in turn depending by the different contribution to output given by each new problem–solving innovation, represented in the model by a vintage of capital. For the last point,the degree of generality of each vintage is a crucial parameter.

Another central assumption of the paper concerns the nature of the firm which, be-sides being a problem–solving entity, is meant here as a carrier of structural and archi-tectural relations between technological components (and labor).

The basic model proposed in the paper is built on many simplifying assumptions; forexample, via their skills workers also embody problem–solving knowledge, and theircapabilities can lead to skill–biased technical change, or to particular diffusion patterns(Nelson and Phelps, 1966), but those features are not modeled. Learning processes takeplace within the firm (Levitt and March, 1988), as well as it happens by doing in theproduction process (Arrow, 1962b). The main motivation of this study is not to denythe traditional sources of incentives for innovative activities (one of which can just be,for example, the capacity of R&D costs spreading for big firms (Cohen, 2010)), but tohighlight what seemed to be a missing piece of the puzzle in the neo–Schumpeterianand evolutionary literature about the determinants of innovation.

In conclusion, firms’ growth (in term of profits and productivity) and heterogene-ity are function of specific and individual capital–vintages–cum–degree–of–generality

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mixes; the model can be extended in order to take into account the market interac-tion between firms, together with the possibility to frame collaborations in innovation(the so-called innovation networks, see Pyka (2002)), imitation and diffusion paths ofproblem-solving innovations. The behavior of macroeconomic aggregates could then beexplained as an emergent property of the micro and meso level non–simple (Simon, 1962)interactions and dynamics.

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